Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(15,21)(16,23)(19,24)(20,25)(30,32)(33,41)(35,39)(37,43);(14,18)(19,24)(20,25)(26,28)(29,30,34,32)(31,35,38,39)(33,40,41,36)(37,44,43,42);(6,10)(7,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,44)(41,43);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,30)(31,35,38,39)(33,40,41,36)(37,42,43,44);(5,8,9,11)(6,7,10,12)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,41)(30,36,32,40)(31,37,38,43)(35,42,39,44)Polycyclic presentation:
2^9|a^2=fghi,b^2=f,c^2=h,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4851309);